June  2020, 19(6): 3209-3222. doi: 10.3934/cpaa.2020139

Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl

Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

* Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: The second author (F.T.) was supported by JSPS Grant-in-Aid for Scientific Research (B), No.19H01800

In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya [4] on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.

Citation: Naoki Hamamoto, Futoshi Takahashi. Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3209-3222. doi: 10.3934/cpaa.2020139
References:
[1]

K. Abe and G. Seregin, Axisymmetric flows in the exterior of a cylinder, to appear in Proceedings of the Royal Society of Edinburgh: Section A Mathematics. doi: 10.1017/prm.2018.121.

[2]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. 

[3]

L. A. CaffarelliR. Kohn and and L. Nirenberg, First order interpolation inequalities with weights, Composito Math., 53 (1984), 259-275. 

[4]

O. Costin and V. Maz'ya, Sharp Hardy-Leray inequality for axisymmetric divergence-free fields, Calc. Var. Partial Differ. Equ., 32 (2008), 523-532.  doi: 10.1007/s00526-007-0151-4.

[5]

C. Efthimiou and C. Frye, Spherical Harmonics in $p$ Dimensions, World Scientific Publishing Company, Singapore, 2014. doi: 10.1142/9134.

[6]

N. Hamamoto, Three-dimensional sharp Hardy-Leray inequality for solenoidal fields, Nonlinear Anal., 191 (2020), Art 111634, 14pp. doi: 10.1016/j.na.2019.111634.

[7]

N. Hamamoto and F. Takahashi, Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields, to appear in Math. Ann. doi: 10.1007/s00208-019-01945-x.

[8] G. H. HardyJ. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952. 
[9]

M. Korobkov, K. Pileckas and R. Russo, The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl, J. Math. Fluid Mech., 17 (2015), 287–293. Addendum: J. Math. Fluid Mech., 18 (2016), 207. doi: 10.1007/s00021-015-0202-0.

[10]

G. KochN. NadirashviliG. Seregin and ">V. , Liouville theorems for the Navier-Stokes equations and applications, Acta. Math., 203 (2009), 83-105.  doi: 10.1007/s11511-009-0039-6.

[11]

O. A. , Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations and applications, Zap. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155–177.

[12]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 12 (1933), 1-82. 

[13]

Y. Liu, and P. Zhang, On the global well-posedness of 3-D axi-symmetric Navier-Stokes system with small swirl component, Calc. Var. Partial Differ. Equ., 57 (2018), 31 pages. doi: 10.1007/s00526-017-1288-4.

[14]

V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition, in Grundlehren der Mathematischen Wissenschaften, Vol. 342, Springer, Heidelberg (2011) xxviii+866. doi: 10.1007/978-3-642-15564-2.

[15]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61.  doi: 10.1016/0021-8928(68)90147-0.

[16]

P. Zhang and T. Zhang, Global axi-symmetric solutions to 3-D Navier-Stokes system, Int. Math. Res. Not. IMRN, 3 (2013), 610-642.  doi: 10.1093/imrn/rns232.

show all references

References:
[1]

K. Abe and G. Seregin, Axisymmetric flows in the exterior of a cylinder, to appear in Proceedings of the Royal Society of Edinburgh: Section A Mathematics. doi: 10.1017/prm.2018.121.

[2]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. 

[3]

L. A. CaffarelliR. Kohn and and L. Nirenberg, First order interpolation inequalities with weights, Composito Math., 53 (1984), 259-275. 

[4]

O. Costin and V. Maz'ya, Sharp Hardy-Leray inequality for axisymmetric divergence-free fields, Calc. Var. Partial Differ. Equ., 32 (2008), 523-532.  doi: 10.1007/s00526-007-0151-4.

[5]

C. Efthimiou and C. Frye, Spherical Harmonics in $p$ Dimensions, World Scientific Publishing Company, Singapore, 2014. doi: 10.1142/9134.

[6]

N. Hamamoto, Three-dimensional sharp Hardy-Leray inequality for solenoidal fields, Nonlinear Anal., 191 (2020), Art 111634, 14pp. doi: 10.1016/j.na.2019.111634.

[7]

N. Hamamoto and F. Takahashi, Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields, to appear in Math. Ann. doi: 10.1007/s00208-019-01945-x.

[8] G. H. HardyJ. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952. 
[9]

M. Korobkov, K. Pileckas and R. Russo, The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl, J. Math. Fluid Mech., 17 (2015), 287–293. Addendum: J. Math. Fluid Mech., 18 (2016), 207. doi: 10.1007/s00021-015-0202-0.

[10]

G. KochN. NadirashviliG. Seregin and ">V. , Liouville theorems for the Navier-Stokes equations and applications, Acta. Math., 203 (2009), 83-105.  doi: 10.1007/s11511-009-0039-6.

[11]

O. A. , Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations and applications, Zap. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155–177.

[12]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 12 (1933), 1-82. 

[13]

Y. Liu, and P. Zhang, On the global well-posedness of 3-D axi-symmetric Navier-Stokes system with small swirl component, Calc. Var. Partial Differ. Equ., 57 (2018), 31 pages. doi: 10.1007/s00526-017-1288-4.

[14]

V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition, in Grundlehren der Mathematischen Wissenschaften, Vol. 342, Springer, Heidelberg (2011) xxviii+866. doi: 10.1007/978-3-642-15564-2.

[15]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61.  doi: 10.1016/0021-8928(68)90147-0.

[16]

P. Zhang and T. Zhang, Global axi-symmetric solutions to 3-D Navier-Stokes system, Int. Math. Res. Not. IMRN, 3 (2013), 610-642.  doi: 10.1093/imrn/rns232.

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